In network information theory, most applications focus on the special case of
(REF ) where one of the random vectors is fixed to be Gaussian. In this
setting, the classical EPI of Shannon and Stam can be further strengthened as
shown by Costa [1]}. Let \(\mathbf {Z}\)  be a Gaussian random \(n\) -vector with
a positive definite covariance matrix, and let \(a\)  be a real scalar such that
\(a \in [0,1]\) . Costa's EPI [1]} can be written as
\(\exp \left[\frac{2}{n}h(\mathbf {X}+\sqrt{a}\mathbf {Z})\right] & \ge (1-a)\exp \left[\frac{2}{n}h(\mathbf {X})\right]+ a\exp \left[\frac{2}{n}h(\mathbf {X}+\mathbf {Z})\right] \) 
